Proof of Black Scholes

Proof of Black Scholes

Mean of a log normal random variable: Theorem 1: Suppose Y = ln X is a normal distribution with mean m and variance v, then X has mean exp( m + v /2 ) Proof: The density function of Y= ln X Therefore the density function of X is given by Using the change of variable x = exp(y), dx = exp(y) dy, We have = Note that the integral inside is just the density function of a normal random variable with mean (m-v) and variance v Your Project Will Be Done Promptly! – click for more info . By definition, the integral evaluates to be 1. Proof of Black Scholes Formula Theorem 2: Assume the stock price following the following PDE

Then the option price for a call option with payoff is given by 1 Proof: By Ito’s lemma, If form a portfolio P Applying Ito’s lemma Since the portfolio has no risk, by no arbitrage, it must earn the risk free rate, Therefore we have Rearranging the terms we have the Black Scholes PDE With the boundary condition To solve this PDE, we need the Feynman-Kac theorem: Assume that f is a solution to the boundary value problem: Then f has the representation: 2 Where S satisfies the following stochastic differential equation

Proof: Suppose that is the solution to the PDE. Let Applying the Ito’s lemma Since the last term involves only second order terms only, Collecting terms we have got As the first term is simply the PDE, it is zero. Therefore Integrating from 0 to T Taking expectation on both side, Since the integral is a limiting sum of independent Brownian motions increments, i. e. =0 it is zero. Recall that W has independent and stationary increment with a zero mean, i. e. is normally distributed with zero mean. 3 Therefore In other words

End of Proof. By the Feynman Kac Theorem, the solution to the Black Scholes PDE is given by Where S follows Consider Z = ln S, by Ito’s lemma, Integrate both side from 0 to T, We have Recall that with mean has a normal distribution with mean 0, and variance T, and variance , let g(X) be the density function of X. has a normal distribution To simplify our notation, let Theorem 3: if ln(X) is a normal distributed random variable and the standard deviation is , then Where 4 Proof: From Theorem 1, the mean of ln X is ,

Define . Q is a standard normal random variable with mean 0 and standard deviation 1. Hence the density function of Q is given by Since pply change of variable Or Let’s consider the first integrand Note that the last expression is nothing more than the density function of a normal random variable with mean and variance 1, i. e. By definition, and apply change of variable again, 5 By definition, the second integrand is End of Proof. Since 1, has a normal distribution with mean and variance , from theorem Applying Theorem 3, Final Exam question 1: Question 1 (total: 25%) Prove the Black Scholes formula a) (2%) If the price of a stock follows the SDE State the pricing formula of the option price for a call option with payoff b) (5%) derive the Black Scholes PDE. c) (5%) State and prove the Feynman Kac theorem. d) (5%) If Y=ln X is a normal distribution with mean m and variance v, show that the mean of X is exp(m +v/2) e) (8%) By applying the Feynman Kac theorem to the Black Scholes PDE, derive the equation you state in part a) 7